| import Mathlib |
| import OAI.Computability.QuantumFactoring.PhasePreparationConnection |
|
|
| namespace OAI |
|
|
| section |
| open scoped BigOperators |
|
|
|
|
| namespace ExactQuantumFactoring |
|
|
| /-- The fixed little-endian physical representation of a modular residue. -/ |
| def basisResidue (b : ℕ) : Basis b ≃ ZMod (2^b) := |
| (basisNumber b).trans (ZMod.finEquiv (2^b)).toEquiv |
|
|
| lemma finEquiv_val {n : ℕ} [NeZero n] (z : Fin n) : |
| (ZMod.finEquiv n z).val = z.val := by |
| cases n with |
| | zero => exact False.elim (NeZero.ne 0 rfl) |
| | succ n => rfl |
|
|
| lemma basisResidue_val (b : ℕ) (x : Basis b) : |
| (basisResidue b x).val = (bitsValue x).toNat := by |
| exact finEquiv_val _ |
|
|
| lemma basisResidue_eq (b : ℕ) (x : Basis b) : |
| basisResidue b x = ((bitsValue x).toNat : ZMod (2^b)) := by |
| rw [← basisResidue_val, ZMod.natCast_zmod_val] |
|
|
| lemma bitsValue_residue (b : ℕ) (z : ZMod (2^b)) : |
| (bitsValue ((basisResidue b).symm z)).toNat = z.val := by |
| rw [← basisResidue_val, Equiv.apply_symm_apply] |
|
|
| lemma basisResidue_bits (b : ℕ) (v : BitVec b) : |
| basisResidue b (fun i => v.getLsbD i.val) = (v.toNat : ZMod (2^b)) := by |
| rw [basisResidue_eq, bitsValue_bits] |
|
|
| lemma bitsValue_residue_add (b : ℕ) (z : ZMod (2^b)) (d : Basis b) : |
| (fun i : Fin b => (bitsValue ((basisResidue b).symm z)+bitsValue d).getLsbD i.val) = |
| (basisResidue b).symm (z+basisResidue b d) := by |
| apply (basisResidue b).injective |
| rw [basisResidue_bits, Equiv.apply_symm_apply, BitVec.toNat_add] |
| rw [ZMod.natCast_mod, Nat.cast_add, bitsValue_residue, ZMod.natCast_zmod_val, |
| basisResidue_eq] |
|
|
| lemma basisResidue_zero (b : ℕ) : basisResidue b (fun _ => false) = 0 := by |
| rw [basisResidue_eq] |
| have h : bitsValue (fun _ : Fin b => false) = (0 : BitVec b) := by |
| apply BitVec.eq_of_getLsbD_eq |
| intro i hi |
| rw [show (bitsValue (fun _ : Fin b => false)).getLsbD i = false from bitsValue_bit _ ⟨i,hi⟩] |
| simp |
| rw [h] |
| simp |
|
|
| end ExactQuantumFactoring |
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|
|
|
| end |
|
|
| end OAI |
|
|
